Object physics not working when moved with armature?
Using an armature to animate a (passive) rigid body using Armature Deform distorts the mesh, rather than actually moving the rigid body - this will affect the final visual state of the mesh but not its underlying location or rotation. The Armature Deform takes place after the simulation and so does not affect the simulation.

In your case you need to parent the 'container' to a specific bone - so that the bone is the Parent of the entire container (rather than using Armature Deform). The container should be a Passive rigid body since its location and rotation will be controlled by the bone, rather than simulated.

By default, the rigid body simulation does not take account of the animated locatation or rotation of the mesh - so the actual influence of the mesh is as if it is still in its original location - this is what you are seeing in your current simulation. However, if you enable the Animated property of the rigid body this tells the simulation engine to take account of any translation and rotation of the mesh before it is used to affect the simulation. This should now result in the contents being affected by the motion of the container, similar to the following :